Physics • Year 12 • Module 8 • Lesson 12
Bohr’s Model of the Hydrogen Atom
Build HSC Band 5–6 extended-response technique on Bohr’s postulates, quantitative spectral analysis, and evaluation of the model’s successes and limitations.
1. Multi-step calculation — Balmer series and photon energy (Band 5–6)
8 marks Band 5–6
Given data: $R_H = 1.097 \times 10^7$ m¹; $h = 6.626 \times 10^{-34}$ J s; $c = 3.00 \times 10^8$ m s¹; $1$ eV $= 1.60 \times 10^{-19}$ J.
Q1. A hydrogen atom emits light when its electron transitions from the fourth energy level ($n = 4$) to the second energy level ($n = 2$), producing the H$\beta$ line of the Balmer series. Answer each part below.
(a) Using the Rydberg formula, calculate the wavelength of the H$\beta$ photon in nm. (2 marks)
(b) Calculate the frequency of the H$\beta$ photon. (1 mark)
(c) Calculate the energy of this photon in both joules and electron-volts. (2 marks)
(d) Verify your answer to (c) by calculating the energy difference $|E_4 - E_2|$ directly from $E_n = -13.6/n^2$ eV. Comment on any discrepancy. (2 marks)
(e) The H$\beta$ line appears blue-green to the human eye. Explain quantitatively why this is consistent with your calculated wavelength. (1 mark)
2. Data + scenario: evaluating Bohr’s model against experiment (Band 5–6)
8 marks Band 5–6
Scenario. In the 1880s, Johann Balmer empirically determined that the visible spectral lines of hydrogen followed the formula $\lambda = B(m^2)/(m^2 - 4)$, where $B = 364.6$ nm and $m = 3, 4, 5, \ldots$ In 1913, Niels Bohr derived a theoretical model with quantised orbits and used it to reproduce Balmer’s results from first principles. The table below compares Bohr-predicted and measured wavelengths for four Balmer lines.
| Line | Transition | Bohr prediction (nm) | Measured value (nm) | Difference (nm) |
|---|---|---|---|---|
| H$\alpha$ | $3 \to 2$ | 656.3 | 656.3 | 0.0 |
| H$\beta$ | $4 \to 2$ | 486.1 | 486.1 | 0.0 |
| H$\gamma$ | $5 \to 2$ | 434.0 | 434.0 | 0.0 |
| H$\delta$ | $6 \to 2$ | 410.2 | 410.2 | 0.0 |
Data based on NIST Atomic Spectra Database. Differences shown to the precision of the values listed.
Q2. Analyse and evaluate the data and the Bohr model to address the following. In your response you must:
- State what the table demonstrates about the agreement between Bohr’s predictions and experiment for hydrogen, using specific values.
- Explain the physical basis of Bohr’s success: what two postulates are most directly responsible for predicting the correct wavelengths?
- Identify two situations where Bohr’s model fails, and for each, briefly explain why it fails.
- Assess whether the success with hydrogen alone is sufficient to regard Bohr’s model as a satisfactory theory of atomic structure. Justify your assessment.
- State one key idea from quantum mechanics (Schrödinger’s model) that resolved one of Bohr’s failures.
Q1(a) — Wavelength of H$\beta$ (2 marks)
$1/\lambda = R_H(1/n_f^2 - 1/n_i^2) = 1.097\times10^7(1/4 - 1/16) = 1.097\times10^7 \times 0.1875 = 2.057\times10^6$ m¹ [1]. $\lambda = 1/(2.057\times10^6) = 4.86\times10^{-7}$ m $= 486$ nm [1].
Marking notes. 1 mark for correct substitution into Rydberg formula with $n_f = 2, n_i = 4$; 1 mark for correct final answer in nm (accept 486 nm ± 1 nm).
Q1(b) — Frequency (1 mark)
$f = c/\lambda = (3.00\times10^8)/(4.86\times10^{-7}) = 6.17\times10^{14}$ Hz [1].
Q1(c) — Photon energy in J and eV (2 marks)
$E = hf = 6.626\times10^{-34} \times 6.17\times10^{14} = 4.09\times10^{-19}$ J [1]. $E = 4.09\times10^{-19}/1.60\times10^{-19} = 2.55$ eV [1].
Marking notes. 1 mark for correct J value; 1 mark for correct eV conversion.
Q1(d) — Verification via energy levels (2 marks)
$E_4 = -13.6/16 = -0.85$ eV; $E_2 = -13.6/4 = -3.40$ eV. $|E_4 - E_2| = |-0.85 - (-3.40)| = 2.55$ eV [1]. This agrees with the value from part (c) to three significant figures. Any small discrepancy arises from rounding in the intermediate step using $\lambda$ (accept 0–0.02 eV difference as rounding) [1].
Marking notes. 1 mark for correctly computing $\Delta E$ from energy levels; 1 mark for noting agreement (or explaining any small rounding discrepancy).
Q1(e) — Colour consistency (1 mark)
Blue-green visible light occupies approximately 480–520 nm. A wavelength of 486 nm falls at the blue-green boundary, consistent with the observed blue-green colour of the H$\beta$ line [1].
Q2 — Sample Band 6 response (8 marks), annotated
Agreement with experiment: The table shows that Bohr’s predictions for all four Balmer lines match the measured wavelengths to four significant figures (e.g. both H$\alpha$ = 656.3 nm, H$\beta$ = 486.1 nm — difference = 0.0 nm in each case). This quantitative agreement was a remarkable achievement in 1913 [1 — cites specific values and notes agreement].
Physical basis of success: Postulate 2 (quantised angular momentum: $L = n\hbar$) gives the specific allowed orbit radii and corresponding energy levels $E_n = -13.6/n^2$ eV. Postulate 3 (photon emitted with $E_{photon} = |E_i - E_f|$) directly links the energy-level differences to the observed photon wavelengths via $E = hc/\lambda$. Together these two postulates are the physical basis for the Rydberg formula and hence the precise wavelength predictions [1 — names Postulate 2 and Postulate 3 with physical content].
Failures of Bohr’s model (two required):
Failure 1 — Multi-electron atoms: Bohr’s model treats only one electron orbiting a nucleus and cannot account for electron–electron repulsion. For helium ($Z=2$, two electrons), the predicted energy levels are significantly wrong because each electron’s orbit is perturbed by the other, an effect that Bohr’s circular-orbit picture cannot incorporate [1].
Failure 2 — Fine structure: High-resolution spectroscopy shows that each spectral line of hydrogen is actually a closely spaced doublet (fine structure). Bohr’s model cannot explain this splitting because it does not include the concept of electron spin or the relativistic correction to electron mass — both of which require quantum mechanics [1]. (Accept also: cannot predict relative intensities of lines; does not explain why angular momentum is quantised.)
Assessment of sufficiency: The success with hydrogen alone is not sufficient to regard Bohr’s model as a satisfactory theory of atomic structure [1 for explicit judgement]. While its quantitative success with hydrogen was historically significant (it was the first model to explain discrete spectral lines from first principles), the model is fundamentally a hybrid of classical and quantum ideas and fails for any system more complex than hydrogen-like ions. A satisfactory atomic theory must explain multi-electron atoms, molecular bonding, and fine structure — requirements that Bohr’s model cannot meet [1 for justification].
Schrödinger’s resolution: Schrödinger’s wave-mechanical model replaced the concept of defined circular orbits with probability wave functions (orbitals). Electrons are described as standing waves around the nucleus; quantisation emerges naturally from boundary conditions on the wave function rather than being imposed arbitrarily as in Bohr’s Postulate 2. This approach successfully extends to multi-electron atoms and explains fine structure via spin quantum numbers [1].
Marking criteria (8 marks): 1 = cites at least two specific wavelength values showing agreement; 1 = names Postulate 2 (quantised angular momentum/orbits) as a basis; 1 = names Postulate 3 (photon emission) as a basis; 1 = failure 1 correctly identified with reason (multi-electron / electron–electron repulsion); 1 = failure 2 correctly identified with reason (fine structure / spin / another valid limitation); 1 = explicit judgement that hydrogen alone is insufficient; 1 = justification of why a general theory must do more; 1 = valid key idea from Schrödinger’s model.